发表机构
Boston College（波士顿学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究高斯RBF再生核希尔伯特空间在大带宽极限下的渐近性质，通过理论分析表明其渐近等距于欧氏空间，核主成分分析相关指标也收敛于线性主成分分析结果，还通过实验证明ρ可预测数据集收敛行为。
AI 中文摘要
我们证明，在大带宽极限下，高斯径向基函数（RBF）再生核希尔伯特空间（RKHS）在各向同性缩放意义下渐近等距于欧几里得空间。这强烈表明，依赖于RKHS度量性质的基于核的构造对于高斯RBF核在大带宽时将产生类似于线性核的结果。高斯CKA的渐近行为也可据此理解。我们进一步考虑核主成分分析，表明随着带宽σ→∞，高斯RBF特征值、特征投影和主成分都收敛于经典（线性）主成分分析的结果。对于给定的数据表示，两种核类型的RKHS特征嵌入和正交主成分分析特征框架在渐近意义下相差一个几何相似变换，余项大小为O((ρ/σ)^2)，其中ρ是表示的几何离心率度量，等于数据示例间最大与中位数成对距离之比。在各种数据集上的实验表明，ρ为特定数据集在顶部主方向上的收敛行为提供了一个简单可靠的预测指标。
英文摘要
We show that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit. This strongly suggests that kernel-based constructions reliant on metric properties of the RKHS will yield results for Gaussian RBF kernels that similarly approach those of linear kernels for large bandwidths. The asymptotic behavior of Gaussian CKA can be understood in this light. We further consider kernel PCA, showing that Gaussian RBF eigenvalues, eigenprojections, and principal components all converge to those of classical (linear) PCA as bandwidth $σ\rightarrow \infty$. For a given data representation, both the RKHS feature embeddings and the orthogonal PCA eigenframes of the two kernel types differ asymptotically by a geometric similarity transformation, up to a residual of size $O \left (\fracρσ \right )^2$, where $ρ$ is a measure of geometric eccentricity of the representation, equal to the ratio of maximum to median pairwise distance between data examples. Experiments over a diverse collection of data sets demonstrate that $ρ$ provides a simple and reliable predictor of dataset-specific convergence behavior in the top principal directions.
